Principal Investigator: Elisabeth Werner
This award supports travel of US participants to a "Workshop on Asymptotic Analysis and Applications" in July 2006 at the Institut Henri Poincare in Paris, France. The workshop is a component of the special trimester research program there on "Phenomena in Large Dimensions" and concerns properties of families of finite-dimensional objects as the dimension grows without bound. Particular topics that fit this agenda are measure transport techniques, concentration of measure, quantitative measures of convex bodies, isoperimetric inequalities, and large deviation inequalities. Some of the important features emerging in these studies are threshold or phase transition effects, similar to those seen in statistical physics or asymptotic combinatorics, and this work also seems to have connections to problems in computational complexity.
A mathematical description of a scientific or engineering question often requires lots of independent numbers, leading to a geometric space of high dimension. For example, if you want to specify the location of one gas molecule in a room then you need to report the front/back, side-to-side, and up/down locations of the molecule, using three numbers. The direction and speed of the molecule's motion takes another three numbers, and so to describe enough of the molecule's current state to allow us to predict its future motion from position and velocity we would need six separate numbers in all. If you want to track 100 distinct molecules of the air in the room then you will need 600 independent numerical coordinates to collect all of the relevant measurements. As these dimensions increase then the difficulty of sampling and computation go up rapidly, a phenomenon scientists and mathematicians sometimes call "the curse of dimensionality." However, there are also patterns that emerge as dimension increases, and this grant for support of US travel to the international workshop described above will study some of these patterns that are recent discoveries.
